Calculus Question?

Question

The following table gives the height h of a rocket in kilometers at a given time t in seconds. (24,12) (27,17) (30,23) (33,31) (36,40) (39,51) With the coordinates being in the form (t,h). Estimate the velocity of the rocket at time = 39 seconds as best you can. Give your answer in km/sec


And Part B for full points of best answer: Using your answer in part a estimate the position of the rocket at time = 40 seconds.

Answer 1

V=at+(V0) and H=1/2 a(t^2) + (V0)t + H0
Note: V0 is the speed at the start of every period of time and H0 is the height at the start of every period of time.

if you calculate the period of time between t=33 and t=39, then
51=1/2*a*[(39-33)^2] + V0*(39-33) + 31 → 20=18*a + 6*V0
if you calculate the period of time between t=33 and t=36, then
40=1/2*a*[(36-33)^2] + V0*(36-33) + 31 → 9=9/2*a + 3*V0
If you solve these 2 formula, you will get the answer: a=2/9=0.222 km/(s^2) and V0=8/3=2.666 km/s
Now you can calculate the speed at time = 39 seconds:
V=at + V0 → V= 2/9 * (39-33) + 8/3 = 6/9 +8/3 = 10/3 = 3.333 km/s
Note: I put (39-33) for t, because V0 is the speed in t=33 and we are going to calculate the speed at time = 39 seconds

Answer 2

If you can use the Matlab software, things are easy:

1) define vector t and h
t=[24,27,30,33,36,39]
h=[12,17,23,31,40,51] 2) find the interpolation polynomial
w= polyfit(t,h,5)
3) for Part B, just put t=40 in the polynomial
b= polyval(w,40)
4) for Part A, find the derivative of the polynomial
v = polyder(t,w)
5) Now find the value of velocity at t=39
velocity =polyval(v,39)

Answer 3

velocity is measured as distance divided by time, so you would need to divide the height of the rocket at 39 seconds to get the velocity.

With this answer, you would apply it to the same formula but would multiply the time by the velocity.

So,

velocity = distance/time

v = 51 km / 39 sec
v = 1.3 km/s

Apply your answer to Part B

v = d / t

d = v x t

d = (1.3 km/s) x (40 s)

d = 52 km

Answer 4

you only have 6 given points....so it must be an equation of maximum 6 unknowns....coefficients of the function h(t)

so h(t) = a t^5+ b t^4+ c t^3+ d t^2+ e t +f


give values of t and find out a,b,c,d,e,f

once found h(t)

velocity = dh/dt = 5a t^4 + 4b t^3 + 3c t^2 + 2d t +e
(replace t with 39)


h(40) = .....